7 research outputs found
Paradigms for Parameterized Enumeration
The aim of the paper is to examine the computational complexity and
algorithmics of enumeration, the task to output all solutions of a given
problem, from the point of view of parameterized complexity. First we define
formally different notions of efficient enumeration in the context of
parameterized complexity. Second we show how different algorithmic paradigms
can be used in order to get parameter-efficient enumeration algorithms in a
number of examples. These paradigms use well-known principles from the design
of parameterized decision as well as enumeration techniques, like for instance
kernelization and self-reducibility. The concept of kernelization, in
particular, leads to a characterization of fixed-parameter tractable
enumeration problems.Comment: Accepted for MFCS 2013; long version of the pape
Don't Be Strict in Local Search!
Local Search is one of the fundamental approaches to combinatorial
optimization and it is used throughout AI. Several local search algorithms are
based on searching the k-exchange neighborhood. This is the set of solutions
that can be obtained from the current solution by exchanging at most k
elements. As a rule of thumb, the larger k is, the better are the chances of
finding an improved solution. However, for inputs of size n, a na\"ive
brute-force search of the k-exchange neighborhood requires n to the power of
O(k) time, which is not practical even for very small values of k.
Fellows et al. (IJCAI 2009) studied whether this brute-force search is
avoidable and gave positive and negative answers for several combinatorial
problems. They used the notion of local search in a strict sense. That is, an
improved solution needs to be found in the k-exchange neighborhood even if a
global optimum can be found efficiently.
In this paper we consider a natural relaxation of local search, called
permissive local search (Marx and Schlotter, IWPEC 2009) and investigate
whether it enhances the domain of tractable inputs. We exemplify this approach
on a fundamental combinatorial problem, Vertex Cover. More precisely, we show
that for a class of inputs, finding an optimum is hard, strict local search is
hard, but permissive local search is tractable.
We carry out this investigation in the framework of parameterized complexity.Comment: (author's self-archived copy
On the (Parameterized) Complexity of Almost Stable Marriage
In the Stable Marriage problem, when the preference lists are complete, all agents of the smaller side can be matched. However, this need not be true when preference lists are incomplete. In most real-life situations, where agents participate in the matching market voluntarily and submit their preferences, it is natural to assume that each agent wants to be matched to someone in his/her preference list as opposed to being unmatched. In light of the Rural Hospital Theorem, we have to relax the "no blocking pair" condition for stable matchings in order to match more agents. In this paper, we study the question of matching more agents with fewest possible blocking edges. In particular, the goal is to find a matching whose size exceeds that of a stable matching in the graph by at least t and has at most k blocking edges. We study this question in the realm of parameterized complexity with respect to several natural parameters, k,t,d, where d is the maximum length of a preference list. Unfortunately, the problem remains intractable even for the combined parameter k+t+d. Thus, we extend our study to the local search variant of this problem, in which we search for a matching that not only fulfills each of the above conditions but is "closest", in terms of its symmetric difference to the given stable matching, and obtain an FPT algorithm
Constraint satisfaction parameterized by solution size
In the constraint satisfaction problem (CSP) corresponding to a constraint
language (i.e., a set of relations) , the goal is to find an assignment
of values to variables so that a given set of constraints specified by
relations from is satisfied. The complexity of this problem has
received substantial amount of attention in the past decade. In this paper we
study the fixed-parameter tractability of constraint satisfaction problems
parameterized by the size of the solution in the following sense: one of the
possible values, say 0, is "free," and the number of variables allowed to take
other, "expensive," values is restricted. A size constraint requires that
exactly variables take nonzero values. We also study a more refined version
of this restriction: a global cardinality constraint prescribes how many
variables have to be assigned each particular value. We study the parameterized
complexity of these types of CSPs where the parameter is the required number
of nonzero variables. As special cases, we can obtain natural and
well-studied parameterized problems such as Independent Set, Vertex Cover,
d-Hitting Set, Biclique, etc.
In the case of constraint languages closed under substitution of constants,
we give a complete characterization of the fixed-parameter tractable cases of
CSPs with size constraints, and we show that all the remaining problems are
W[1]-hard. For CSPs with cardinality constraints, we obtain a similar
classification, but for some of the problems we are only able to show that they
are Biclique-hard. The exact parameterized complexity of the Biclique problem
is a notorious open problem, although it is believed to be W[1]-hard.Comment: To appear in SICOMP. Conference version in ICALP 201